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The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The '''generalised continuum hypothesis''' (GCH) states that <math>\beth_{a} = \aleph_{a}</math> for all ordinals <math>a</math>. So, the continuum hypothesis is a special case of the generalised continuum hypothesis for ordinal <math>0</math>. Like CH, GCH is independent of the axioms of ZFC.

The '''generalized continuum hypothesis''' (GCH) states that <math>\beth_{a} = \aleph_{a}</math> for all ordinals <math>a</math>. So, the continuum hypothesis is a special case of the generalised continuum hypothesis for ordinal <math>0</math>. Like CH, GCH is independent of the axioms of ZFC.

A set having cardinality equal to or less than <math>\aleph_{0}</math> is called countable; a set of greater cardinality is called uncountable.

A set having cardinality equal to or less than <math>\aleph_{0}</math> is called countable; a set of greater cardinality is called uncountable.